Finite Group Databasebeta

← #38746009

Subgroups — #38746009

Order 25216.

432 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C197⋊#401C211yesmaximal, characteristic, complemented
2#27161021C211yesmaximal, characteristic, complemented
3#27161020C211yesmaximal, characteristic, complemented
4#27161017C211yesmaximal, characteristic, complemented
5#27161007C211yesmaximal, characteristic
6#27161007C211yesmaximal, characteristic
7#27161002C211yesmaximal, characteristic
8C4⋊C4×Dic197C211yesmaximal, characteristic, complemented
9#27160930C211yesmaximal, characteristic, complemented
10#27160930C211yesmaximal, characteristic, complemented
11C22⋊C4×Dic197C211yesmaximal, characteristic
12C22⋊C4×Dic197C211yesmaximal, characteristic
13C22⋊C4×Dic197C211yesmaximal, characteristic
14#27160924C211yesmaximal, characteristic, complemented
15#397×C197C211yesmaximal, characteristic
16C23×Dic197C2211yescharacteristic
17C2×C4⋊Dic197C2211yescharacteristic
18C2×#15691337C2211yescharacteristic
19C2×#15691337C2211yescharacteristic
20C2×#15691337C2211yescharacteristic
21C2×#15691337C2211yescharacteristic
22C2×C22⋊Dic197C2211yescharacteristic
23C2×C22⋊Dic197C2211yescharacteristic
24C2×C22⋊Dic197C2211yescharacteristic
25C2×C4×Dic197C2211yescharacteristic
26C2×C4×Dic197C2211yescharacteristic
27C2×C4×Dic197C2211yescharacteristic
28C2×C4×Dic197C2211yescharacteristic
29C2×C4×Dic197C2211yescharacteristic
30C2×C4×Dic197C2211yescharacteristic
31#20263233C2211yescharacteristic
32#20263233C2211yescharacteristic
33#20263233C2211yescharacteristic
34#20263233C2211yescharacteristic
35#20263233C2211yescharacteristic
36#20263233C2211yescharacteristic
37#20263233C2211yescharacteristic
38#20263233C2211yescharacteristic
39#20263233C2211yescharacteristic
40#20263233C2211yescharacteristic
41#20263233C2211yescharacteristic
42#20263233C2211yescharacteristic
43#20263233C2211yescharacteristic
44C2×C4⋊C4×C197C2211yescharacteristic
45C2×C22⋊C4×C197C2211yescharacteristic
46C2×C22⋊C4×C197C2211yescharacteristic
47C2×C22⋊C4×C197C2211yescharacteristic
48#95×C197C2211yescharacteristic
49#95×C197C2211yescharacteristic
50#95×C197C2211yescharacteristic
51C22×Dic197—22no—
52C22×Dic197—22no—
53C22×Dic197—22no—
54C22×Dic197C2311yescharacteristic
55C22×Dic197C2311yescharacteristic
56C22×Dic197C2311yescharacteristic
57C22×Dic197C2311yescharacteristic
58C22×Dic197C2311yescharacteristic
59C22×Dic197C2311yescharacteristic
60C22×Dic197C2311yescharacteristic
61C22×Dic197C2311yescharacteristic
62C4⋊Dic197—22no—
63#15691337—22no—
64#15691337—22no—
65#15691337—22no—
66#15691337—22no—
67C22⋊Dic197—22no—
68C22⋊Dic197—22no—
69C22⋊Dic197—22no—
70C4×Dic197—22no—
71C4×Dic197—22no—
72C4×Dic197—22no—
73C4×Dic197—22no—
74C4×Dic197—22no—
75C4×Dic197—22no—
76C24×C197C2311yescharacteristic
77C22×C4×C197C2311yescharacteristic
78C22×C4×C197C2311yescharacteristic
79C22×C4×C197C2311yescharacteristic
80C22×C4×C197C2311yescharacteristic
81C22×C4×C197C2311yescharacteristic
82C22×C4×C197C2311yescharacteristic
83C4⋊C4×C197—22no—
84C22⋊C4×C197—22no—
85C22⋊C4×C197—22no—
86C22⋊C4×C197—22no—
87C2×Dic197—21no—
88C2×Dic197—21no—
89C2×Dic197—21no—
90C2×Dic197—21no—
91C2×Dic197—21no—
92C2×Dic197—21no—
93C2×Dic197—21no—
94C2×Dic197—21no—
95C2×Dic197—21no—
96C2×Dic197—21no—
97C2×Dic197—21no—
98C2×Dic197—21no—
99C2×Dic197—21no—
100C2×Dic197—21no—
101C2×Dic197—21no—
102C2×Dic197—21no—
103C2×Dic197—21no—
104C2×Dic197—21no—
105C2×Dic197—21no—
106C2×Dic197—44no—
107C2×Dic197#4112yes—
108C2×Dic197#4112yes—
109C2×Dic197#4112yes—
110C2×Dic197#4112yes—
111C2×Dic197#4112yes—
112C23×C197—21no—
113C23×C197—21no—
114C23×C197—21no—
115C23×C197—21no—
116C23×C197—21no—
117C23×C197—21no—
118C23×C197#4112yes—
119C23×C197C2411yescharacteristic
120C2×C4×C197—21no—
121C2×C4×C197—21no—
122C2×C4×C197—21no—
123C2×C4×C197—21no—
124C2×C4×C197—21no—
125C2×C4×C197—21no—
126C2×C4×C197—21no—
127C2×C4×C197—21no—
128C2×C4×C197—21no—
129C2×C4×C197—21no—
130C2×C4×C197—21no—
131C2×C4×C197—21no—
132C2×C4×C197—21no—
133C2×C4×C197—21no—
134C2×C4×C197—21no—
135C2×C4×C197—21no—
136C2×C4×C197—21no—
137C2×C4×C197#4112yes—
138C22×C197—22no—
139C22×C197—22no—
140C22×C197—22no—
141C22×C197—41no—
142C22×C197—41no—
143C22×C197—41no—
144C22×C197—41no—
145C22×C197C2×#4111yescharacteristic
146C22×C197C2×#4111yescharacteristic
147C22×C197C2×#4111yescharacteristic
148C22×C197C2×#4111yescharacteristic
149C22×C197C2×#4111yescharacteristic
150C22×C197C2×#4111yescharacteristic
151C22×C197C2×#4111yescharacteristic
152C4×C197—22no—
153C4×C197—41no—
154C4×C197—41no—
155C4×C197—41no—
156C4×C197—41no—
157C4×C197—41no—
158Dic197—22no—
159Dic197—22no—
160Dic197—41no—
161Dic197—41no—
162Dic197—41no—
163Dic197—41no—
164Dic197—41no—
165Dic197#12614yes—
166C2×C197—42no—
167C2×C197C2×#12611yescharacteristic
168C2×C197#52911yescharacteristic
169C2×C197#52911yescharacteristic
170C2×C197#52911yescharacteristic
171C2×C197#52911yescharacteristic
172C2×C197#52911yescharacteristic
173C2×C197#52911yescharacteristic
174C197#240111yescharacteristic, complemented, Sylow
175#2401—1971nomaximal, Sylow
176#401—1971no—
177#400—1971no—
178#400—1971no—
179#400—1971no—
180#397—1971no—
181#397—1971no—
182#397D19711yescharacteristic
183#396—1971no—
184C4×C4⋊C4—1971no—
185C4×C4⋊C4—1971no—
186C4×C4⋊C4—1971no—
187C4×C22⋊C4—1971no—
188C4×C22⋊C4—1971no—
189C4×C22⋊C4—1971no—
190#376—1971no—
191C23×C4—1971no—
192C2×C4⋊C4—1971no—
193C2×C4⋊C4—1971no—
194C2×C4⋊C4—1971no—
195C2×C4⋊C4—1971no—
196C2×C4⋊C4—1971no—
197C2×C4⋊C4C2×D19711yescharacteristic
198C2×C22⋊C4—1971no—
199C2×C22⋊C4—1971no—
200C2×C22⋊C4—1971no—
201C2×C22⋊C4C2×D19711yescharacteristic
202C2×C22⋊C4C2×D19711yescharacteristic
203C2×C22⋊C4C2×D19711yescharacteristic
204C2×C42—1971no—
205C2×C42—1971no—
206C2×C42—1971no—
207C2×C42—1971no—
208C2×C42—1971no—
209C2×C42—1971no—
210#95—1971no—
211#95—1971no—
212#95—1971no—
213#95—1971no—
214#95—1971no—
215#95—1971no—
216#95—1971no—
217#95—1971no—
218#95—1971no—
219#95—1971no—
220#95—1971no—
221#95—1971no—
222#95—1971no—
223#95C2×D19711yescharacteristic
224#95C2×D19711yescharacteristic
225#95C2×D19711yescharacteristic
226C24C22×D19711yescharacteristic
227C22×C4—3942no—
228C22×C4—3942no—
229C22×C4—3942no—
230C22×C4—1971no—
231C22×C4—1971no—
232C22×C4—1971no—
233C22×C4—1971no—
234C22×C4—1971no—
235C22×C4—1971no—
236C22×C4—1971no—
237C22×C4—1971no—
238C22×C4C22×D19711yescharacteristic
239C22×C4C22×D19711yescharacteristic
240C22×C4C22×D19711yescharacteristic
241C22×C4C22×D19711yescharacteristic
242C22×C4C22×D19711yescharacteristic
243C22×C4C22×D19711yescharacteristic
244C4⋊C4—22no—
245C4⋊C4—3942no—
246C4⋊C4—3942no—
247C4⋊C4—3942no—
248C4⋊C4—3942no—
249C4⋊C4—3942no—
250C22⋊C4—22no—
251C22⋊C4—22no—
252C22⋊C4—22no—
253C22⋊C4—3942no—
254C22⋊C4—3942no—
255C22⋊C4—3942no—
256C42—3942no—
257C42—3942no—
258C42—3942no—
259C42—3942no—
260C42—3942no—
261C42—3942no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1569135712yes—
269C23C23×D19711yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—7884no—
288C2×C4—3941no—
289C2×C4—3941no—
290C2×C4—3941no—
291C2×C4—3941no—
292C2×C4—3941no—
293C2×C4—3941no—
294C2×C4—3941no—
295C2×C4—3941no—
296C2×C4—3941no—
297C2×C4—3941no—
298C2×C4—3941no—
299C2×C4—3941no—
300C2×C4—3941no—
301C2×C4—3941no—
302C2×C4—3941no—
303C2×C4—3941no—
304C2×C4—3941no—
305C2×C4—3941no—
306C2×C4—3941no—
307C2×C4—1972no—
308C2×C4—1972no—
309C2×C4—1972no—
310C2×C4—1972no—
311C2×C4—1972no—
312C2×C4#1569135712yes—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—41no—
317C22—41no—
318C22—41no—
319C22—41no—
320C22#41×D19711yescharacteristic
321C22#41×D19711yescharacteristic
322C22#41×D19711yescharacteristic
323C22#41×D19711yescharacteristic
324C22#41×D19711yescharacteristic
325C22C2×#1569135711yescharacteristic
326C22C2×#1569135711yescharacteristic
327C4—22no—
328C4—41no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—7881no—
334C4—7881no—
335C4—7881no—
336C4—7881no—
337C4—7881no—
338C4—3942no—
339C4—3942no—
340C4—1974no—
341C2—42no—
342C2#126×D19711yescharacteristic
343C2#2716160611yescharacteristic
344C2#2716160611yescharacteristic
345C2#2716160611yescharacteristic
346C2#2716160811yescharacteristic
347C2#2716161011yescharacteristic
348C2#2716161211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D197
commutator, socle, cosocleC23×C197C24
Fitting#397×C197C2
prime radicalC24×C197C23
hypercenter#397D197
hypocenterC197#2401
LCSG ▷ C23×C197 ▷ C197
UCSC1 ◁ C23 ◁ #397

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C197⋊#401C2semidirect
2. #27161021C2semidirect
3. #27161020C2semidirect
4. #27161017C2semidirect
8. C4⋊C4×Dic197C2semidirect
9. #27160930C2semidirect
10. #27160930C2semidirect
14. #27160924C2semidirect
174. C197#2401semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C197⋊#4012
#271610212
#271610202
#271610172
#271610072
#271610022
C4⋊C4×Dic1972
#271609302
C22⋊C4×Dic1972
#271609242
#397×C1972
#2401197